25 tone equal temperament

25EDO divides the octave in 25 equal steps of exact size 48 cents each. It is a good way to tune the Blackwood temperament, which takes the very sharp fifths of 5EDO as a given, tempers out 28/27 and 49/48, and attempts to optimize the tunings for 5 (5/4) and 7 (7/4). It also tunes sixix temperament with a sharp fifth. It supplies the optimal patent val for the 11-limit 6&25 temperament tempering out 49/48, 77/75 and 605/576, and the 13-limit extension also tempering out 66/65.

25EDO has fifths 18 cents sharp, but its major thirds are excellent and its 7/4 is acceptable. Moreover, in full 7-limit including the 3, it is not consistent. It therefore makes sense to use it as a 2.5.7 subgroup tuning. Looking just at 2, 5, and 7, it equates five 8/7s with the octave, and so tempers out (8/7)^5 / 2 = 16807/16384. It also equates a 128/125 diesis and two septimal tritones of 7/5 with the octave, and hence tempers out 3136/3125. If we want to temper out both of these and also have decent fifths, the obvious solution is 50EDO. An alternative fifth, 14\25, which is 672 cents, provides an alternative very flat fifth which can be used for mavila temperament.

If 5/4 and 7/4 aren't good enough, it also does 17/16 and 19/16, just like 12EDO. In fact, on the 2*25 subgroup 2.9.5.7.33.39.17.19 it provides the same tuning and tempers out the same commas as 50et, which makes for a wide range of harmony.

Music

Study in Fives by Paul Rapoport
Fantasy for Piano in 25 Note per Octave Tuning play by Chris Vaisvil
Flat fourth blues by Fabrizio Fulvio Fausto Fiale

25edochorale.mid

Peter Kosmorsky (10/14/10, 2.5.7 subgroup, a friend responded "The 25edo canon has a nice theme, but all the harmonizations from there are laughably dissonant. I showed them to my roomie and he found it disturbing, hahaha. He had an unintentional physical reaction to it with his mouth in which his muscles did a smirk sort of thing, without him even trying to, hahaha. So, my point; this I think this 25 edo idea was an example of where tonal thinking doesn't suit the sound of the scale.")

25 edo prelude largo.mid

Peter Kosmorsky (2011, Blackwood)

Intervals

Degrees	Cents	Approximate Ratios*	Armodue Notation	ups and downs notation
0	0	1/1	1	P1	perfect 1sn	D, Eb
1	48	33/32, 39/38, 34/33	1#	^1, ^m2	up 1sn, upminor 2nd	D^, Eb^
2	96	17/16, 20/19, 18/17	2b	^^m2	double-upminor 2nd	Eb^^
3	144	12/11, 38/35	2	vvM2	double-downmajor 2nd	Evv
4	192	9/8, 10/9, 19/17	2#	vM2	downmajor 2nd	Ev
5·	240	8/7	3b	M2, m3	major 2nd, minor 3rd	E, F
6	288	19/16, 20/17	3	^m3	upminor 3rd	F^
7	336	39/32, 17/14, 40/33	3#	^^m3	double-upminor 3rd	F^^
8·	384	5/4	4b	vvM3	double-downmajor 3rd	F#vv
9	432	9/7, 32/25, 50/39	4	vM3	downmajor	F#v
10	480	33/25, 25/19	4#/5b	M3, P4	major 3rd, perfect 4th	F#, G
11·	528	31/21, 34/25	5	^4	up 4th	G^
12	576	7/5, 39/28	5#	^^4,^^d5	double-up 4th, double-up dim 5th	G^^, Ab^^
13	624	10/7, 56/39	6b	vvA4,vv5	double-down aug 4th, double-down 5th	G#vv, Avv
14·	672	42/31, 25/17	6	v5	down 5th	Av
15	720	50/33, 38/25	6#	P5, m6	perfect 5th, minor 6th	A, Bb
16	768	14/9, 25/16, 39/25	7b	^m6	upminor 6th	Bb^
17·	816	8/5	7	^^m6	double-upminor 6th	Bb^^
18	864	64/39, 28/17, 33/20	7#	vvM6	double-downmajor 6th	Bvv
19	912	32/19, 17/10	8b	vM6	downmajor 6th	Bv
20·	960	7/4	8	M6, m7	major 6th, minor 7th	B, C
21	1008	16/9, 9/5, 34/19	8#	^m7	upminor 7th	C^
22	1056	11/6, 35/19	9b	^^m7	double-upminor 7th	C^^
23	1104	32/17, 17/9, 19/10	9	vvM7	double-downmajor 7th	C#vv
24	1152	33/17, 64/33, 76/39	9#/1b	vM7	downmajor 7th	C#v
25	1200	2/1	1	P8	perfect 8ve	C#, D

*based on treating 25-EDO as a 2.9.5.7.33.39.17.19 subgroup; other approaches are possible.

25ed2-001.svg

Relationship to Armodue

Like 16-EDO and 23-EDO, 25-EDO contains the 9-note "Superdiatonic" scale of 7L2s (LLLsLLLLs) that is generated by a circle of heavily-flattened 3/2s (ranging in size from 5\9-EDO or 666.67 cents, to 4\7-EDO or 685.71 cents). The 25-EDO generator for this scale is the 672-cent interval. This allows 25-EDO to be used with the Armodue notation system in much the same way that 19-EDO is used with the standard diatonic notation; see the above interval chart for the Armodue names. Because the 25-EDO Armodue 6th is flatter than that of 16-EDO (the middle of the Armodue spectrum), sharps are lower in pitch than enharmonic flats.

Commas

25 EDO tempers out the following commas. (Note: This assumes the val < 25 40 58 70 86 93 |.)

Comma	Monzo	Value (Cents)	Name 1	Name 2	Name 3
256/243	\| 8 -5 >	90.22	Limma	Pythagorean Minor 2nd
3125/3072	\| -10 -1 5 >	29.61	Small Diesis	Magic Comma
	\| 38 -2 -15 >	1.38	Hemithirds Comma
16807/16384	\| -14 0 0 5 >	44.13
49/48	\| -4 -1 0 2 >	35.70	Slendro Diesis
64/63	\| 6 -2 0 -1 >	27.26	Septimal Comma	Archytas' Comma	Leipziger Komma
3125/3087	\| 0 -2 5 -3 >	21.18	Gariboh
50421/50000	\| -4 1 -5 5 >	14.52	Trimyna
1029/1024	\| -10 1 0 3 >	8.43	Gamelisma
3136/3125	\| 6 0 -5 2 >	6.08	Hemimean
65625/65536	\| -16 1 5 1 >	2.35	Horwell
100/99	\| 2 -2 2 0 -1 >	17.40	Ptolemisma
176/175	\| 4 0 -2 -1 1 >	9.86	Valinorsma
91/90	\| -1 -2 -1 1 0 1 >	19.13	Superleap
676/675	\| 2 -3 -2 0 0 2 >	2.56	Parizeksma